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A totally different method was incorporated by Andreas Erdmann into
the lithography simulator SOLID [144,146,147].
As this approach is based on the finitedifference beam propagation
method [148] it is not restricted to laterally homogeneous resists.
However, it is only suited for a planar topography. The field calculation is
based on a numerical solution of the Helmholtz equation
(cf. (4.2)) that writes inside the inhomogeneous resist as

(5.30) 
whereby a transversalelectric polarized light and a twodimensional simulation
domain is required [144]. The field amplitude
E_{y, k}(x, z) is
separated into a slowly varying amplitude A_{k}(x, z) obeying
(/z^{2})A_{k}(x, z) = 0 and
an exponential factor
exp(
jk_{0}z) describing the propagation upwards
and downwards the resist, i.e.,

(5.31) 
Insertion of (5.32) into (5.31)
transforms the Helmholtz equation into

(5.32) 
which for example can be solved on an equidistant grid with a CrankNicholson
scheme [144]. The boundary conditions can either be
transparent ones [149] used in case of isolated features, or periodic
ones applied for dense lines and spaces.
Consideration of reflective substrates requires
additional modifications as either downward or upward propagationbut not
simultaneouslycan be modeled by (5.32).
However, reflections occurring at the air/resist
as well as the resist/substrate interface can conveniently be calculated by
Fresnel's reflection formulae [11, pp. 3651].
The light is then repeatedly propagated down and up through the resist until its
intensity is negligible or, alternatively, until a fixed number of iterations
is performed and then simpler methods, e.g., the transfer matrix algorithm,
are employed.
In the above described form the beam propagation is only suited for
low
NA applications as the beams are assumed to travel almost
parallel to the vertical axis. An extended wide angle algorithm
exists [150] that is suited for higher numerical aperture lithography
simulation. Numerical problems such as poor convergence and instability of the
solution occur in case of a strongly varying refractive index.
A more rigorous finitedifference time domain propagation method is then
required [151]. The main limitation of this method is its restriction
to planar layers. In the next section we describe various methods suited
for nonplanar topography.
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Up: 5.2 Field Calculation over
Previous: 5.2.3 Transfer Matrix Method
Heinrich Kirchauer, Institute for Microelectronics, TU Vienna
19980417